Digital Systems Number Systems and Codes

Digital Systems
Number Systems and Codes
Wen-Hung Liao, Ph.D.
Objectives
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Convert a number from one number system (decimal, binary, octal,
hexadecimal) to its equivalent in one of the other number systems.
Cite the advantages of the octal and hexadecimal number
systems.
Count in octal and hexadecimal.
Represent decimal numbers using the BCD code; cite the pros
and cons of using BCD.
Understand the difference between BCD and straight binary.
Understand the purpose of alphanumeric codes such as the ASCII
code.
Explain the parity method for error detection.
Determine the parity bit to be attached to a digital data string
Binary-to-Decimal Conversions
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Example 1: 110112
Example 2: 101101012
Decimal-to-Binary Conversions
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Method one: reverse the process of binary-todecimal conversion.
Method two: repeated division
Example: 3710=1001012
Octal Number System
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The octal number system has a base of eight.
Eight possible digits: 0,1,2,3,4,5,6,7
Octal point
Octal-to-decimal conversion: 3728
Decimal-to-octal conversion: 26610
Octal-to-binary conversion
Binary-to-octal conversion
Octal system can be used as a “shorthand” for
expressing large binary numbers.
Hexadecimal Number System
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The hexadecimal number system has a base
of 16.
Sixteen possible digits:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Hex-to-decimal conversion
Decimal-to-hex conversion
Hex-to-binary conversion
Binary-to-hex conversion
BCD Code
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Binary-Coded-Decimal versus straight binary
coding.
0  0000, 1 0001, 20010, 30011,
40100, 50101, 60110, 70111, 8
1000, 9 1001
874 (decimal) 1000 0111 0100 (BCD)
Alphanumeric Codes
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ASCII code: American Standard Code for
Information Interchange
The ASCII code is a 7 bit code, so it has
2^7=128 possible code groups.
Refer to Table 2-4.
Parity Method for Error Detection
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Whenever information is transmitted from one device to
another device, errors can occur due to noise.
Parity method can be used to detect error.
A parity bit is an extra bit that is attached to a code
group that is being transferred.
In even-parity method, the value of the parity bit is
chosen so that the total # of 1s in the code group
(including the parity bit) is an even number.
In odd-parity method, the value of the parity bit is
chosen so that the total # of 1s in the code group
(including the parity bit) is an odd number.
Example
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ASCII ‘C’: 1000011
Even-parity method: 1 1000011
Odd-parity method: 0 1000011
The parity bit is issued to detect any single-bit
errors that occur during the transmission